# Worst-case evaluation complexity and optimality of second-order methods for nonconvex smooth optimization

arXiv: https://arxiv.org/abs/1709.07180 · HTML: https://mlsanity.com/paper/1709.07180/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Sep 2017.

## Built on by (most cited first, 6 of 6)

- [Deterministic Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/2302.08300/) (2023)
- [Gradient Shaping: Enhancing Backdoor Attack Against Reverse Engineering](https://mlsanity.com/paper/2301.12318/) (2023)
- [A Newton-CG based barrier method for finding a second-order stationary point of nonconvex conic optimization with complexity guarantees](https://mlsanity.com/paper/2207.05697/) (2022)
- [Sharp First-Order Lower Bounds for Higher-Order Smooth Nonconvex Optimization](https://mlsanity.com/paper/2606.05438/) (2026)
- [Newton-CG methods for nonconvex unconstrained optimization with H"older continuous Hessian](https://mlsanity.com/paper/2311.13094/) (2023)
- [The Row Normalization Puzzle in Muon](https://mlsanity.com/paper/2609.39114/) (2026)

## Abstract (arXiv)

We establish or refute the optimality of inexact second-order methods for unconstrained nonconvex optimization from the point of view of worst-case evaluation complexity, improving and generalizing the results of Cartis, Gould and Toint (2010,2011). To this aim, we consider a new general class of inexact second-order algorithms for unconstrained optimization that includes regularization and trust-region variations of Newton's method as well as of their linesearch variants. For each method in this class and arbitrary accuracy threshold ε ∈ (0,1), we exhibit a smooth objective function with bounded range, whose gradient is globally Lipschitz continuous and whose Hessian is α-H"older continuous (for given α∈ [0,1]), for which the method in question takes at least lfloorε-(2+α)/(1+α)rfloor function evaluations to generate a first iterate whose gradient is smaller than ε in norm. Moreover, we also construct another function on which Newton's takes lfloorε-2rfloor evaluations, but whose Hessian is Lipschitz continuous on the path of iterates. These examples provide lower bounds on the worst-case evaluation complexity of methods in our class when applied to smooth problems satisfying the relevant assumptions. Furthermore, for α=1, this lower bound is of the same order in ε as the upper bound on the worst-case evaluation complexity of the cubic and other methods in a class of methods proposed in Curtis, Robinson and samadi (2017) or in Royer and Wright (2017), thus implying that these methods have optimal worst-case evaluation complexity within a wider class of second-order methods, and that Newton's method is suboptimal.